Reflections In A Spiral Mirror

The title of this post looks like it could be the name of a concept album by a pretentious prog-rock band. But it’s completely literal—I came across the spiral mirror in question while walking back from Tralee into Benderloch the other day. It was an outdoor ornament of the kind that seems to be called a “spiral wind twister”, and it was conveniently dangling from a sign beside the road.I had a camera in my pocket, so I had the chance to photograph a puzzling little phenomenon that was first pointed out to me by Dave Hewitt and Chris Tyler.

If you look closely, there are two odd things about my reflection in the spiral mirror of the wind twister—one is that I’m turned sideways; the other is (as you can see from the readable text on the sign behind me) that I’m not mirror-reversed. A mirror that rotates but doesn’t reverse! I don’t know about you, but I certainly didn’t see that one coming.

It must be something to do with the complex shape of the reflective surface. It takes a moment to tease this out, but the spiral produces a mirror that is convex on one axis and concave on another. The two axes are roughly at right angles to each other, and at forty-five degrees to the vertical. I’ve marked the concave axis in red on the enlargement below—you can see the edges of the mirror curling to face you at the top and bottom of that line. The convex axis is in green, with the mirror curving away from you in both directions along that axis.

To start working out what’s going on, I’m going to look at examples of mirrors that are either purely convex or purely concave. Here’s a photo of the cover of a book reflected in a rather spiffy Venetian convex mirror:The convexity lends a fish-eye distortion to what is otherwise just a conventional mirror-image reflection. There’s nothing new going on there, which suggests we can ignore the convex component of the spiral mirror when we try to tease out the cause of its remarkable reflection.

Now here’s the same book reflected in a concave shaving mirror, with the (zoomed) photograph taken from a couple of metres away.As you can see, concave mirrors do something interesting. When you’re close to them they act as magnifying mirrors. But if you step back outside the focus of the mirror, they turn into inverting mirrors.

This is what’s going on:The curve of the mirror means you need to look up to see the reflection of your feet, and down to see the reflection of your face. Likewise, you need to look left to see your right hand, and right to see your left hand. In effect, a uniformly concave mirror flips the standard mirror image both left-to-right and top-to-bottom. Below, I’ve marked those rotation axes with black lines superimposed on a standard mirror image of the book:

These two reflections turn out to be the equivalent of a 180° rotation—the image in the concave mirror is still mirror-reversed, but now it’s upside-down, too.

But the wind twister was concave in only one direction. Suppose we had a mirror that was only concave from top-to-bottom. There are metal mirrors like that behind the elements in old-fashioned two-bar electric heaters, like this ancient example I lugged out of the attic for illustrative purposes:

A cylindrical concave mirror like that will generate only a top-to-bottom image flip, and not a right-to-left flip:Ah-ha! The top-to-bottom flip, combined with the usual mirror-reversal, gives us an inverted but unreversed final image. Now we’re getting someplace.

Finally, we just need to remember that the concave curve of the wind twister is orientated diagonally, so it’s going to do a diagonal flip on the reflected image. Like this:Ta-da! I’ve finally reconstructed the 90°-rotated but unreversed image from the wind twister. Who’d have thought you get so much out of peering closely at a garden ornament?

Note: There’s an old puzzle: Why does a mirror reverse left and right but not top and bottom? I’ve skated around that, above, by talking only about “the usual mirror reversal”. But, actually, a mirror doesn’t reverse left and right at all—it reverses front and back. The person you look at in the mirror has head, feet, left hand and right hand in the same places as you do, but is facing in the opposite direction. It’s keeping left and right in the same place while reversing front and back that turns your reflection into a mirror image.